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\title{Property about $\cuu(\fgg)^K$ action}

\def\propk{{property $K$ }}
\def\A{{\mathcal{A}}}
\def\S{{\mathcal{S}}}
\def\ugk{{\cuu(\fgg)^K}}
\def\zkk{\mathcal{Z}(\fkk)}
\begin{document}
\maketitle

\begin{dfn}
We say a set $\S$ of representation has \propk
 if there exist a algebra $\A$ and map 
$\phi\colon \ugk \to \A$  
such that the $\ugk$ action  on 
space $\Hom_K(V,W)$ (as a right module) factor through $\phi$ where $V\in \S$ and $W$ 
is any irreducible representation of $K$
and $\zkk$ surjective to $\A$.
\end{dfn}

\begin{rmk}
By noticing the following commutative diagram
\[

\]
\end{rmk}


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